Valuation overrings of a Noetherian domain

Abstract

Let R be a Noetherian domain and 0=P0⊊P1⊊⋯⊊Pn be a saturated chain of prime ideals of R. Let V be a valuation overring of R that has a chain of prime ideals {Qα}α∈Λ such that {Qα∩R}α∈Λ={Pi}i=0n. In this paper, we prove that {Qα}α∈Λ={0=Q0⊊Q1⊊⋯⊊Qn} and VQn is discrete, i.e., QiVQi is principal for all i=1,…,n. Let D be an integral domain such that DP is Noetherian for each prime ideal P of D with htP<∞. As a corollary, we show that if {Pk} is a chain of prime ideals of D such that htPk<∞ for each k, then there exists a discrete valuation overring of D which has a chain of prime ideals lying over {Pk}.